Finite Group Databasebeta

← #10877228

Subgroups — #10877228

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75965C211yesmaximal, characteristic
2D3×#384C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4#69612C211yesmaximal, characteristic, complemented
5#69590C211yesmaximal, characteristic, complemented
6#68904C212yesmaximal, complemented
7#68878C212yesmaximal
8#68874C212yesmaximal
9#68795C212yesmaximal, complemented
10#68691C211yesmaximal, characteristic, complemented
11#68662C211yesmaximal, characteristic
12#27681—31nomaximal, Sylow
13C2×#897C2211yescharacteristic
14C2×D3×C4⋊C4—21no—
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4C2212yes—
17C2×#887C2211yescharacteristic
18C2×C42×D3C2211yescharacteristic
19C3×#389C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×(C2×D3)⋊C8C2212yes—
22C2×#818C2212yes—
23C2×#816C2212yes—
24C2×Dic3⋊C8C2212yes—
25#4989—21no—
26#4987C2211yescharacteristic
27#4975—21no—
28#4975—21no—
29#4975C2212yes—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4C2211yescharacteristic
32C2×#810C2212yes—
33C2×#807C2212yes—
34#4935C2211yescharacteristic
35#4929C2211yescharacteristic
36#4666—21no—
37#4654—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C22⋊Q8—31no—
45C2×#384—31no—
46#2100—31no—
47#2100D311yescharacteristic, complemented
48#2095—31no—
49#2095—31no—
50#1947—32no—
51#1872—32no—
52#1852—32no—
53#1803—32no—
54#1797—31no—
55C22×C4×D3—22no—
56C22×C4×D3D411yescharacteristic
57C22×C4×D3D411yescharacteristic
58C22×C4×D3C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85#897—22no—
86#897Q814yes—
87D3×C4⋊C4—42no—
88D3×C4⋊C4—42no—
89D3×C4⋊C4—44no—
90D3×C4⋊C4—44no—
91#887—24no—
92C42×D3—24no—
93C3×#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98(C2×D3)⋊C8—24no—
99#818—42no—
100#816—42no—
101Dic3⋊C8—24no—
102#810—42no—
103#807—42no—
104C2×C22⋊Q8—31no—
105C2×C4×Q8—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—61no—
108C22×C4⋊C4—61no—
109C22×C42—31no—
110C2×#107—32no—
111C2×#107—32no—
112C2×C4⋊C8—32no—
113C2×#103—32no—
114C2×#103—32no—
115C2×C22⋊C8—32no—
116#391—31no—
117#389C2×D311yescharacteristic
118#387—61no—
119#385—61no—
120#384—32no—
121#384—31no—
122#384—61no—
123#384—61no—
124#384C2×D311yescharacteristic
125#380—31no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—61no—
128C4×C22⋊C4—61no—
129#340—32no—
130#340—32no—
131#340C2×D312yes—
132#340C2×D312yes—
133#336—31no—
134#336C2×D311yescharacteristic
135C23×D3C2×D411yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3C2×D411yescharacteristic
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—44no—
158C2×C4×D3Q1614yes—
159C2×C4×D3C2×Q812yes—
160C2×C4×D3#4112yes—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—21no—
164C2×Dic6C2×Q812yes—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4C2×Q812yes—
176C3×C42—22no—
177(C2×D3)⋊C4—24no—
178(C2×D3)⋊C4—42no—
179C4⋊Dic3—24no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—42no—
184C4⋊Dic3—42no—
185Dic3⋊C4—44no—
186Dic3⋊C4—44no—
187Dic3⋊C4—42no—
188Dic3⋊C4—44no—
189C4×Dic3—22no—
190C4×Dic3—22no—
191C4×Dic3—42no—
192C4×Dic3—42no—
193C4×Dic3#4114yes—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C22×Q8—31no—
198C23×C4—31no—
199C23×C4—31no—
200C23×C4—31no—
201C23×C4—62no—
202C22×C8—32no—
203C22×C8C22×D312yes—
204C22⋊Q8—34no—
205C22⋊Q8—62no—
206C4×Q8—64no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—122no—
220C2×C4⋊C4—122no—
221C2×C4⋊C4—124no—
222C2×C4⋊C4—124no—
223C2×C4⋊C4C22×D312yes—
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C42—31no—
229C2×C42—31no—
230C2×C42—31no—
231C2×C42—64no—
232C2×C42—61no—
233C2×C42—61no—
234C2×C42C22×D311yescharacteristic
235#107—122no—
236#107—122no—
237C4⋊C8—64no—
238#103—122no—
239#103—122no—
240C22⋊C8—64no—
241#95—21no—
242#95—61no—
243#95—61no—
244#95—61no—
245#95—61no—
246C23×C3C22×D411yescharacteristic
247C22×D3—21no—
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3C22⋊Q812yes—
252C22×D3C2×Q1612yes—
253C22×D3AGL(1,ℤ/8)12yes—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4C2×Q1612yes—
268C2×C3×C4#13712yes—
269C2×C3×C4C22×Q811yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4112yes—
272C2×C3×C4H(5,𝔽2)12yes—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—22no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—41no—
286C2×Dic3C22⋊Q812yes—
287C2×Dic3#12312yes—
288C2×Dic3#12312yes—
289C2×Dic3C2×Q1612yes—
290C2×Dic3AGL(1,ℤ/8)12yes—
291C4×D3—28no—
292C4×D3—24no—
293C4×D3—24no—
294C4×D3—28no—
295C4×D3—88no—
296Dic6—24no—
297Dic6—42no—
298C3×C8—42no—
299C3⋊C8—42no—
300C24—31no—
301C2×Q8—32no—
302C2×Q8—61no—
303C2×Q8—62no—
304C2×Q8—62no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—34no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—124no—
325C22×C4—124no—
326C22×C4—124no—
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4C23×D311yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—64no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—62no—
347C4⋊C4—122no—
348C4⋊C4—122no—
349C4⋊C4—124no—
350C4⋊C4—122no—
351C4⋊C4—124no—
352C4⋊C4—124no—
353C4⋊C4D3×Q812yes—
354C22⋊C4—64no—
355C22⋊C4—122no—
356C42—22no—
357C42—34no—
358C42—62no—
359C42—62no—
360C42—122no—
361C42—122no—
362C22×C3C2×C22⋊Q811yescharacteristic
363C22×C3C2×#12311yescharacteristic
364C22×C3#53511yescharacteristic
365C22×C3C22×Q1611yescharacteristic
366C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
367C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
368C22×C3C2×#13711yescharacteristic
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3#47714yes—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4#55312yes—
382C3×C4#55412yes—
383Dic3—22no—
384Dic3—22no—
385Dic3—42no—
386Dic3—82no—
387Dic3—81no—
388Dic3#48414yes—
389C23—32no—
390C23—32no—
391C23—32no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23C2×D3×D411yescharacteristic
397Q8—64no—
398Q8—122no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—68no—
418C2×C4—68no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—248no—
435C2×C4#91512yes—
436C2×C4D3×Q1612yes—
437C2×C4C2×D3×Q811yescharacteristic
438C2×C4D3×#4112yes—
439C2×C4D3×#4111yescharacteristic
440C2×C4#100912yes—
441C8—42no—
442C8—122no—
443C2×C3#248711yescharacteristic
444C2×C3C2×#47711yescharacteristic
445C2×C3#307911yescharacteristic
446C2×C3C2×#48411yescharacteristic
447C2×C3#309211yescharacteristic
448C2×C3#320911yescharacteristic
449C2×C3#321711yescharacteristic
450D3—24no—
451C22—34no—
452C22—62no—
453C22—62no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22D3×C22⋊Q811yescharacteristic
459C22D3×#12311yescharacteristic
460C22#567411yescharacteristic
461C22C2×#91511yescharacteristic
462C22C2×D3×Q1611yescharacteristic
463C22D3×AGL(1,ℤ/8)11yescharacteristic
464C22D3×AGL(1,ℤ/8)11yescharacteristic
465C4—22no—
466C4—22no—
467C4—34no—
468C4—42no—
469C4—62no—
470C4—62no—
471C4—82no—
472C4—81no—
473C4—122no—
474C4—242no—
475C4—241no—
476C4#569012yes—
477C4#570912yes—
478C3#2768111yescharacteristic, complemented, Sylow
479C2—64no—
480C2#7814611yescharacteristic
481C2#8106111yescharacteristic
482C2D3×#47711yescharacteristic
483C2#8132511yescharacteristic
484C2D3×#48411yescharacteristic
485C2#8160411yescharacteristic
486C2#8171711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2100C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2100D3
hypocenterC3#27681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#2100C2semidirect
4. #69612C2semidirect
5. #69590C2semidirect
6. #68904C2semidirect
9. #68795C2semidirect
10. #68691C2semidirect
47. #2100D3semidirect
478. C3#27681semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#759652
D3×#3842
C3×#21002
#696122
#695902
#689042
#688782
#688742
#687952
#686912
#686622
#276813